Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 175 · 40 questions

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

1. 1.Which is the correct factorisation of x² + 17x + 72? (a) (x − 8)(x − 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x + 8)(x + 9)
2. 2.Using (a + b)² = a² + 2ab + b², what is 74²? (a) 4916 (b) 5486 (c) 4356 (d) 5476
3. 3.Using (a − b)² = a² − 2ab + b², what is 99²? (a) 9801 (b) 9999 (c) 10201 (d) 9811
4. 4.Using (a + b)² = a² + 2ab + b², what is 52²? (a) 2704 (b) 2304 (c) 2714 (d) 2504
5. 5.Which is the correct factorisation of x² + 15x + 56? (a) (x + 1)(x + 14) (b) (x + 7)(x + 8) (c) (x + 1)(x + 56) (d) (x − 7)(x − 8)
6. 6.If a + b = 4 and ab = 3, what is a² + b²? (a) 22 (b) 16 (c) 13 (d) 10
7. 7.Which is the correct factorisation of x² + 5x + 6? (a) (x + 2)(x + 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x − 2)(x − 3)
8. 8.What is the sum of the roots of x² − 12x + 35 = 0? (a) 35 (b) 12 (c) 13 (d) 11
9. 9.What are the roots of x² − 13x + 42 = 0? (a) 2 and 21 (b) 1 and 12 (c) 6 and 7 (d) 1 and 42
10. 10.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x − 3)(x − 5) (c) (x + 3)(x + 5) (d) (x + 1)(x + 7)
11. 11.If a + b = 13 and ab = 36, what is (a − b)²? (a) 169 (b) 5 (c) 97 (d) 25
12. 12.What is the product of the roots of x² − 12x + 35 = 0? (a) 36 (b) 35 (c) 12 (d) 34
13. 13.If a + b = 17 and ab = 60, what is (a − b)²? (a) 169 (b) 49 (c) 7 (d) 289
14. 14.Using (a − b)² = a² − 2ab + b², what is 18²? (a) 484 (b) 396 (c) 324 (d) 334
15. 15.If a + b = 12 and ab = 35, what is a² + b²? (a) 214 (b) 109 (c) 74 (d) 144
16. 16.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 961 (b) 899 (c) 851 (d) 841
17. 17.What is the sum of the roots of x² − 14x + 48 = 0? (a) 14 (b) 13 (c) 48 (d) 15
18. 18.Using (a − b)² = a² − 2ab + b², what is 97²? (a) 9419 (b) 9409 (c) 9991 (d) 10609
19. 19.Which is the correct factorisation of x² + 11x + 30? (a) (x + 1)(x + 10) (b) (x + 1)(x + 30) (c) (x − 5)(x − 6) (d) (x + 5)(x + 6)
20. 20.Which is the correct factorisation of x² + 7x + 12? (a) (x + 1)(x + 12) (b) (x + 1)(x + 6) (c) (x − 3)(x − 4) (d) (x + 3)(x + 4)
21. 21.What is the sum of the roots of x² − 12x + 27 = 0? (a) 13 (b) 11 (c) 27 (d) 12
22. 22.Using (a − b)² = a² − 2ab + b², what is 19²? (a) 399 (b) 441 (c) 361 (d) 371
23. 23.What is the product of the roots of x² − 8x + 15 = 0? (a) 16 (b) 14 (c) 15 (d) 8
24. 24.What are the roots of x² − 12x + 35 = 0? (a) 1 and 11 (b) 5 and 7 (c) 2 and 10 (d) 1 and 35
25. 25.If a + b = 12 and ab = 20, what is a² + b²? (a) 184 (b) 124 (c) 144 (d) 104
26. 26.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 433 (c) 145 (d) 289
27. 27.Using (a + b)² = a² + 2ab + b², what is 44²? (a) 1616 (b) 1946 (c) 1936 (d) 1296
28. 28.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 324 (d) 2
29. 29.If a + b = 17 and ab = 70, what is (a − b)²? (a) 3 (b) 149 (c) 9 (d) 289
30. 30.If a + b = 6 and ab = 5, what is (a − b)²? (a) 16 (b) 4 (c) 26 (d) 36
31. 31.Using (a − b)² = a² − 2ab + b², what is 49²? (a) 2601 (b) 2499 (c) 2401 (d) 2411
32. 32.Using (a + b)² = a² + 2ab + b², what is 84²? (a) 6416 (b) 7056 (c) 7066 (d) 5776
33. 33.What are the roots of x² − 13x + 36 = 0? (a) 1 and 12 (b) 1 and 36 (c) 4 and 9 (d) 2 and 18
34. 34.Using (a + b)² = a² + 2ab + b², what is 24²? (a) 586 (b) 416 (c) 256 (d) 576
35. 35.Using (a − b)² = a² − 2ab + b², what is 16²? (a) 384 (b) 266 (c) 256 (d) 576
36. 36.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 12) (b) (x + 6)(x + 7) (c) (x − 6)(x − 7) (d) (x + 1)(x + 42)
37. 37.What are the roots of x² − 15x + 56 = 0? (a) 7 and 8 (b) 2 and 28 (c) 1 and 14 (d) 1 and 56
38. 38.Which is the correct factorisation of x² + 11x + 28? (a) (x + 1)(x + 28) (b) (x + 4)(x + 7) (c) (x + 1)(x + 10) (d) (x − 4)(x − 7)
39. 39.If a + b = 4 and ab = 3, what is (a − b)²? (a) 4 (b) 2 (c) 16 (d) 10
40. 40.What is the product of the roots of x² − 7x + 12 = 0? (a) 7 (b) 13 (c) 11 (d) 12

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